Fractional multiplicative processes

www.imstat.org/aihp 2009, Vol. 45, No. 4, 1116–1129

DOI: 10.1214/08-AIHP198

© Association des Publications de l’Institut Henri Poincaré, 2009

Fractional multiplicative processes

Julien Barral

and Benoît Mandelbrot

aINRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. E-mail: [email protected] bYale University, Mathematics Department, New Haven CT 06520, USA. E-mail: [email protected]

Received 12 February 2008; revised 29 October 2008

Abstract. Statistically self-similar measures on[0,1]are limit of multiplicative cascades of random weights distributed on the b-adic subintervals of[0,1]. These weights are i.i.d., positive, and of expectation 1/b. We extend these cascades naturally by allowing the random weights to take negative values. This yields martingales taking values in the space of continuous functions on [0,1]. Specifically, we consider for eachH∈(0,1)the martingale(Bn)_n≥1obtained when the weights take the values−b^−H and b^−H, in order to getBnconverging almost surely uniformly to a statistically self-similar functionBwhose Hölder regularity and fractal properties are comparable with that of the fractional Brownian motion of exponentH. This indeed holds whenH∈(1/2,1).

Also the construction introduces a new kind of law, one that it is stable under random weighted averaging and satisfies the same functional equation as the standard symmetric stable law of index 1/H. WhenH ∈(0,1/2], to the contrary,Bndiverges almost surely. However, a natural normalization factora_nmakes the normalized correlated random walkB_n/a_n converge in law, asn tends to∞, to the restriction to[0,1]of the standard Brownian motion. Limit theorems are also associated with the caseH >1/2.

Résumé. Les mesures sur[0,1]auto-similaires en loi sont limites de processus multiplicatifs construits à partir de poids aléatoires distribués sur les sous-intervallesb-adiques de[0,1]. Ces poids sont i.i.d., positifs et d’espérance 1/b. Il est naturel d’étendre la construction à des poids prenant des valeurs négatives. On obtient alors des martingales à valeurs dans les fonctions continues sur [0,1]. Nous nous intéressons, pourH∈(0,1), à la martingale(Bn)_n≥1de ce type construite en prenant des poids à valeurs dans {−b^−H, b^−H}, afin queB_nconverge presque sûrement uniformément vers une fonctionBauto-similaire en loi dont la régularité Höldérienne et les propriétés fractales soient comparables à celles du mouvement brownien fractionnaire d’exposantH. C’est bien le cas lorsqueH∈(1/2,1), et la construction fournit alors un nouvel exemple de loi invariante par moyenne pondérée aléatoire.

Cette loi satisfait la même équation fonctionnelle qu’une loi stable symétrique usuelle d’indice 1/H. SiH∈(0,1/2],Bndiverge presque sûrement, mais il existe une normalisation naturelle par une suite(an)_n≥1telle que la marche aléatoire corrélée normalisée B_n/a_nconverge en loi vers la restriction à[0,1]du mouvement brownien standard. Des théorèmes limites sont également associés au casH >1/2.

MSC:Primary: 60F05; 60F15; 60F17; 60G18; 60G42; secondary 28A78

Keywords:Random functions; Martingales; Central Limit Theorem; Brownian motion; Laws stable under random weighted mean; Fractals;

Hausdorff dimension

1. Introduction and results

Measure-valued martingales associated with cascades were introduced in [24,25] as a “canonical” model for intermit- tent turbulence. They are generated by multiplicative cascades of positive random weights distributed on the nodes of a homogeneous tree. When non-degenerate, these martingales converge to singular multifractal measures whose fine study has led to numerous developments, both in probability and geometric measure theories (see [1–4,7,9,12,14,15, 18,19,24,26,27]). We consider the natural extension of these martingales consisting in allowing the random weights to take negative values.

We simplify the exposition by using cascades in basis 2 (the necessary complements to extend our results in basis b≥3 are given in Remark 1.4). The dyadic closed subintervals of[0,1]are naturally encoded by the nodes of the binary treeT =

n≥0{0,1}ⁿ, with the convention that{0,1}⁰contains the root ofT denoted∅. As in the definition of positive canonical cascades [24], we associate to each elementwofT a real valued random weightW (w); these weights are i.i.d. andE(W )is defined and equal to 1/2. A sequence of random continuous piecewise linear functions (Bn)n≥1is then obtained as follows:Bn(0)=0;Bnis linear over every dyadic intervalI of thenth generation; ifI is encoded by the nodew1w2· · ·wn, i.e.I=Iw:= [n

k=1wk2^−k,2⁻ⁿ+n

k=1wk2^−k], the increment ofBnoverI is the productW (w1)W (w1w2)· · ·W (w1w2· · ·w_n). IfW is non-negative, the derivatives in the distributions sense of the functionsB_nform the measure-valued martingale considered in [19,24,25].

This paper investigates the signed cascades in which the weightW takes the same absolute value throughout, in order to generate fractional Brownian motion (fBm) like processes (see [21,23] for the definition of fBm). It is not difficult to see that in this case, for some H∈(−∞,1],W must be of the formW=2^−H, whereis a random variable taking the values 1 and−1 with respective probabilitiesp⁺=(1+2^H−1)/2 andp⁻=(1−2^H−1)/2. Then let us reformulate the definition of(B_n)_n≥1.

Consider a sequence((w))_w∈T of independent copies ofand for everyn≥1 andw=w₁· · ·w_n∈ {0,1}ⁿdefine (w)=

n k=1

(w1· · ·w_k)∈ {−1,1}. (1.1)

We can writeB_nas a normalized correlated random walk as follows: Forn≥1 and 0≤k <2ⁿ defineξ_k⁽ⁿ⁾=(w), wherew=w1· · ·w_nis the unique element of{0,1}ⁿsuch thatt_w=n

i=1w_i2⁻ⁱ=k2⁻ⁿ. The random variablesξ_k⁽ⁿ⁾, 0≤k <2ⁿ, are identically distributed and they take values in{−1,1}. Also, consider the random walk

S_r⁽ⁿ⁾=

r−1

k=0

ξ_k⁽ⁿ⁾, 0≤r <2ⁿ

(with the conventionS₋⁽ⁿ⁾₁=0). Then fort∈ [0,1]we have Bn(t )=2^−nH

S_[⁽ⁿ⁾₂nt]+ 2ⁿt−

2ⁿt ξ_[⁽ⁿ⁾₂nt]

. (1.2)

An equivalent definition of(Bn)n≥1is B_n(t )=2^−nH

_t

0

2ⁿ(u₁)· · ·(u₁· · ·u_n)du,

where the sequence (uk)k≥1 stands for the digits of uin basis 2. This second definition shows by inspection that this sequence of random continuous functions forms a martingale with respect to the filtration(Fn)n≥1, whereFn= σ{(w): w∈_n

k=1{0,1}^k}.

For everyp≥0 andw=w1· · ·w_p∈ {0,1}^pwe consider the copy of(B_n)_n≥1defined by B_n(w)(t )=2^−nH

_t

0

2ⁿ(w·u₁)· · ·(w·u₁· · ·u_n)du, n≥1,

wherew·u₁· · ·u_kis the concatenation of the wordswandu₁· · ·u_k. By construction,B_n(∅)=B_nand the following stochastic scaling invariance holds. With probability 1, for alln≥1 andt∈Iw

Bp+n(t )−Bp+n(tw)=(w)2^−pHBn(w)

S_w⁻_p¹◦ · · · ◦S_w⁻₁¹(t ) , (1.3) whereS0(t )=t /2 andS1(t )=(t+1)/2.

The previous properties of Bn may seem to suggest that ifH ∈(0,1), the construction provides a simple way to generate a sequence of normalized random walks (see (1.2)) converging almost surely uniformly to a functionB

Fig. 1. Bkfork=8,12,18,27 in the caseb=2 andH=0.95: Fast strong convergence.

possessing scaling and fractal properties close to those of a fBm of exponentH. In fact, our study of(B_n)_n≥1shows the situation to be subtler and heavily dependent onH, a kind of phase transition arising atH=1/2.

WhenH∈(1/2,1), the martingale(Bn)n≥1indeed converges as expected asntends to∞(Theorem 1.1). This is illustrated in Figs 1 and 2. The pointwise Hölder exponent of the almost sure limitBis equal toH everywhere, and the Hausdorff dimension of the graph ofBis 2−H. Moreover, the processBpossesses scaling invariance properties relative to the dyadic grid, withH playing the role of a Hurst exponent, as can be seen by lettingntend to∞in (1.3).

Furthermore, the normalized processB/

E(B(1)²)converges in law to the standard Brownian motion asH1/2 (Theorem 1.2). Thus,B shares a lot of properties with fBm of exponentH, though it has not stationary increments and it is not Gaussian (see Remark 1.1). WhenH ∈(−∞,1/2], the martingale is not bounded inL²norm and it diverges. However, the normalized sequenceB_n/

E(B_n(1)²)converges in law to the standard Brownian motion as ntends to∞(Theorem 1.3). This is illustrated in Figs 3 and 4. WhenH <1/2 this result is a version of Donsker’s theorem, but for triangular arrays with unusual strong correlations. WhenH=1/2, the same strong correlations hold, butB_n/

E(B_n(1)²)corresponds to a correlated random walk normalized in the same unusual way as very different correlated random walks considered in [11] and weakly converging to Brownian motion as well (see the discussion in Remark 1.3).

Our results are stated and commented in the following theorems and remarks. Then we relate them with some works on laws that are stable under random weighted mean.

C([0,1])will denote the space of real-valued continuous functions over[0,1]endowed with the uniform norm denoted by · _∞, and Id_[0,1] will denote the identity function over[0,1]. We refer to [13] for the definitions of Hausdorff and box dimensions of sets inR^das well as [6] for the theory of the convergence of probability measures on metric spaces.

The caseH∈(1/2,1].

Theorem 1.1. LetH∈(1/2,1].TheC([0,1])-valued martingale(B_n)_n≥1 converges almost surely and inL^q norm for allq≥1to a limit function of expectationId_[0,1].Denote this limit byBand for allw∈T the limit ofB_n(w)by B(w).With probability1,

1. For allp≥1,w∈ {0,1}^pandt∈Iw

B(t )−B(t_w)=(w)2^−pHB(w) S_w⁻¹

p◦ · · · ◦S_w⁻¹

1(t ); (1.4)

Fig. 2. B_kfork=8,12,18,27 in the caseb=2 andH=0.7. Strong convergence.

Fig. 3. B_k/σ√

kfork=8,12,18,27 in the caseb=2 andH=0.5: Convergence in distribution to the Wierner Brownian motion.

2. B isα-Hölder continuous for allα∈(0, H ),and it has everywhere a pointwise Hölder exponent equal toH,i.e.

for allt∈ [0,1]

lim inf

s→t s=t

log|B(s)−B(t )| log|s−t| =H;

3. The Hausdorff and box dimensions of the graph ofBare equal to2−H. ForH∈(1/2,1)defineσ_H =(2−2^2−2H)^−1/2=

E(B(1)²)(this equality will be justified in the proof of the next result) and denoteB byB_H.

Theorem 1.2. The family of continuous processes{B_H/σ_H}H∈(1/2,1) converges in law, asH tends to1/2,to the restriction to[0,1]of the standard Brownian motion.

Remark 1.1. WhenH=1,the weights are positive and the construction coincides with the trivial positive cascade:

with probability1, Bn(t )=t for allt ∈ [0,1] and n≥1. WhenH ∈(1/2,1),the limit process B−Id_[0,1] is not

Fig. 4. Bk/σ b^{k(1/2−H )}fork=8,12,18,27 in the caseb=2 andH= −2: Convergence in distribution to the Wierner Brownian motion.

fractional Brownian motion.This can be seen on(1.4)since(w)is not symmetric.Also,a computation shows that the third moment of the centered random variableB(1)−1does not vanish,so the process is not Gaussian.

The caseH∈ [−∞,1/2]

ForH∈(−∞,1/2], the sequence (B_n)_n≥1 is not bounded in L² norm. To get a natural normalization making it bounded inL²norm let

σ= 1+

2^2−2H−2 ⁻¹ ifH <1/2, 1/√

2 ifH=1/2

and forw∈T andn≥1 define X_n(w)=

B_n(w)/σ2^{n(1/2−H )} ifH <1/2, B_n(w)/σ√

n ifH=1/2.

Also simply denoteXn(∅)byXn. The processXnis equivalent toBn/

E(Bn(1)²)asntends to∞(this fact will be justified in the proof of the next result). If we letH tend to −∞in the definition ofandσ, then becomes a symmetric random variable taking values in{−1,1},σ=1, and the sequence(X_n)_n≥1has the natural extension to the caseH= −∞given byX_n(t )=^√1_2n[S_[⁽ⁿ⁾₂nt]+(2ⁿt− [2ⁿt])ξ_[⁽ⁿ⁾₂nt]](see Remark 1.3).

Theorem 1.3. For everyH∈ [−∞,1/2]the sequence of continuous processes(Xn)n≥1converges in law,asntends to∞,to the restriction to[0,1]of the standard Brownian motion.

Remark 1.2. WhenH ∈(−∞,1/2),Theorem1.3yields lim sup_n→∞B_n∞2^{−n(1/2−H )}>0 almost surely.Thus the martingale(B_n)_n≥1diverges in C([0,1]).The same property holds when H=1/2. Besides,Theorem1.1says that(B_n)_n≥1 converges almost surely uniformly to a limit of expectation Id_[0,1] whenH >1/2.Consequently,the convergence properties of non-positive canonical cascades strongly depend on the random weight used to generate the process.This contrasts with the positive canonical cascades martingales,which always converge almost surely uniformly(either to a non-trivial limit with expectation Id_[0,1],or to0, see [19,24]).

Remark 1.3. WhenH∈(−∞,1/2],due to(1.2)we have X_n(t )=

⎧⎨

⎩

1 σ√ 2ⁿ

S_[⁽ⁿ⁾₂nt]+ 2ⁿt−

2ⁿt ξ_[⁽ⁿ⁾₂nt]

ifH <1/2,

1 σ√

n2ⁿ

S_[⁽ⁿ⁾₂nt]+ 2ⁿt−

2ⁿt ξ_[⁽ⁿ⁾₂nt]

ifH=1/2. (1.5)

WhenH <1/2, the form ofX_nis familiar from Donsker’s theorem (see [6]) and its extensions to triangular arrays of random variables that are weakly dependent (see [6,8]). However, the correlations of the X_n dyadic increments are closely related to the natural ultrametric distance onT and it seems difficult to find a way to reduce the behavior of(X_n)_n≥1to that of random walks with weakly dependent increments. WhenH=1/2, theX_n dyadic increments are correlated as well, and the normalization of the random walk is similar to the unusual one met in the proof of Theorem 2 in [11] (see also Lemma 5.1 of [28]) to obtain the weak convergence to Brownian motion of certain centered stationary Gaussian random walks.

If we denoteX_n(w)(1)byY_n(w), the relation (1.7) below yields

Yn+1=

⎧⎨

⎩

(0)√

2Y_n(0)+^(1)√₂Y_n(1) ifH <1/2, n

n+1

₍₀₎

√2Y_n(0)+^(1)√₂Y_n(1) ifH=1/2. (1.6)

Consequently, assuming thatX_nconverges in law, we can guess thanks to (1.6) that the weak limit ofY_nmust be the standard normal distribution. Actually, to identify this limit we exploit the recursive equations (1.6) as well as recursive equations satisfied by the moments of the standard normal distribution (see (3.1) in the proof of Lemma 3.1). A similar approach exploiting the functional Eq. (2.2) is used to prove Theorem 1.2.

LettingH tend to−∞yieldsσ=1 and a random variablethat takes the values−1 and 1 with equal probability 1/2 so that the random walkS_r⁽ⁿ⁾becomes symmetric. In this case, the convergence in law to Brownian motion of X_n(defined as in (1.5) in the limitH= −∞) follows from standard arguments, sinceX_n conditioned with respect toGn−1=σ{(w): w∈ {0,1}ⁿ⁻¹}satisfies the Donsker’s theorem assumptions (givenGn−1, theξ_k⁽ⁿ⁾s are symmetric, independent, and take values−1 and 1).

IfH∈(1/2,1)andσis defined asσ=

E(B(1)²)−1, the same kind of argument can be used to prove thatX_n= (B−B_n)/σ2^{n(1/2−H )}also converges in law to Brownian motion. Indeed, due to (1.4), conditionally onσ{(w): w∈ {0,1}ⁿ}, the increments of the process 2^n/2Xnover the dyadic intervals of generationnare 2ⁿindependent centered random variables distributed like(B(1)−1)/σ or−(B(1)−1)/σ, namely the(w)(B(w)(1)−1)/σ,w∈ {0,1}ⁿ, whose standard deviation is equal to 1.

A link with laws that are stable under random weighted mean

Forn≥0 andw∈T we denote byZ_n(w)the random variable B_n(w)(1), with the conventionB₀(w)(1)=1. We simply writeZ_nforZ_n(∅). By construction, for everyn≥1

Z_n=2^−H(0)Zn−1(0)+2^−H(1)Zn−1(1), (1.7)

where the random variables(0),(1),Z_n−1(0)andZ_n−1(1)are mutually independent,(0)and(1)are copies of, andZn−1(0)andZn−1(1)are copies ofZn−1. Relation (1.7) is central in the sequel. When the martingale(Zn)n≥1

does converge to a non trivial limitZ(see Theorem 1.1), it follows from (1.7) that the probability distribution ofZ provides a new family of what has been called law stable by random weighted mean or fixed points of the smoothing transformation ([9,14,24]). Indeed, there exist two independent copiesZ(0)andZ(1)ofZ, and two independent and identically distributed random variablesW (0)andW (1)– namely, 2^−H(0)and 2^−H(1)– such that(W (0), W (1)) is independent of(Z(0), Z(1))andZsatisfies the following equality in distribution (≡)

Z≡W (0)Z(0)+W (1)Z(1). (1.8)

When (W (0), W (1)) is positive, the non-trivial positive solutions of this equation are described in [9,14,19,24].

A class of non-positive solutions of (1.8) with positive (W (0), W (1)) has been exhibited in [22]; it naturally in- cludes classical symmetric stable laws of index α∈ [1,2], which obey (1.8) when W (0)=W (1)=2^−H with

H =1/α∈ [1/2,1]. Actually, the classical symmetric stable law of index α=1/H ∈ [1,2] satisfies Eq. (1.8) under the form Z≡2^−Hη(0)Z(0)+2^−Hη(1)Z(1) as soon as η(0) and η(1) are independent, take values −1 and 1, and are independent of (Z(0), Z(1)), whatever be the distributions of η(0) andη(1). Consequently, when (η(0), η(1))=((0), (1)), Theorem 1.1 provides for eachH∈(1/2,1]another probability distribution obeying the same functional equation as the classical symmetric stable law of index 1/H. It is worth noting that the statistically self-similar stochastic processes associated with these solutions have very different behaviors. In the first case, if H=1/α∈(1/2,1]the process is a symmetric stable Lévy processL_α of indexα(see [5]), so the distributions of the increments have no finite moments of order larger than or equal toα, and the sample path ofL_αhave a dense set of discontinuities and are multifractal [17]. In the second case, the process is the random functionB of Theorem 1.1, the distributions of the dyadic increments have a finite moment of orderpfor allp >0, and the sample path ofBare continuous and monofractal.

Remark 1.4. Both the construction and results extend to the case when the construction grid isb-adic withb≥3.

Then W =b^−H, where =1 with probability (1+b^H−1)/2 and = −1 with probability(1−b^H−1)/2. The same results hold after formal replacement of the basis2 by the basisb. Also, σ =√

1−1/b if H =1/2, σ = 1+(b−1)/(b^2−2H−b)ifH <1/2,andσ_H=√

b−1/√

b−b^2−2H ifH >1/2.

Theorems 1.1–1.3 are proved in Sections 2–4 respectively.

2. Proof of Theorem 1.1

Lemma 2.1. The martingale(Z_n=B_n(1))n≥1converges almost surely and inL^qnorm for allq≥1.

Proof. For every integerq≥1, raising (1.7) to the powerqyields E

Z_n^q₊₁ =2^1−qHE ^q E

Z^q_n +2^{−H q}

q−1

k=1

q k

E

^k E

^q−k E Z^k_n E

Z_n^q−k . (2.1)

Moreover, sinceH >1/2 we have 0<2^1−qHE(^q) <1 for all integersq≥2 (E(^q)is equal to 2^H−1ifqis odd and 1 otherwise). Consequently, sinceE(Z_n)=1 for alln≥1, induction on q∈N^∗using (2.1) shows that the sequence E(Z^q_n)converges asntends to∞for every integerq≥1. This implies that the martingale(Z_n)_n≥1is bounded inL^q

norm for allq≥1, hence the result.

Lemma 2.2. Letα∈(0, H ).With probability1,there exists an integerp0≥1such that

∀p≥p₀, sup

0≤k≤2^p−1

sup

n≥1

B_n

(k+1)2^−p −B_n

k2^−p ≤2^−pα.

Proof. For everyp≥1 and 0≤k≤2^p−1, the sequence (ΔB_n(p, k)=B_n((k+1)2^−p)−B_n(k2^−p))_n≥1 is by construction a martingale, so Doob’s inequality yields for everyq >1 a constantC_q>0 such that

E supn≥1

ΔBn(p, k)^q

≤Cqsup

n≥1

EΔBn(p, k)^q .

On the one hand – by construction – ifn≤p, thenE(|ΔB_n(p, k)|^q)=2^−qn(H−1)2^−qp≤2^−qpH. On the other hand, (1.3) and Lemma 2.1 together yield a constantC_q≥1 such thatE(|ΔB_n(p, k)|^q)≤C_q2^−qpH ifn > p. Consequently, for allp≥1,

E supn≥1

ΔBn(p, k)^q

≤CqC_q2^−qpH.

Forq > (H−α)⁻¹, the previous inequality implies

p≥1

P

∃0≤k <2^p: sup

n≥1

ΔB_n(p, k)>2^−pα

<∞.

We conclude thanks to the Borel–Cantelli lemma.

Forw∈T we defineZ(w)=lim_n→∞Z_n(w), and we denoteZbyZ(∅).

Lemma 2.3. Letϕstand for the characteristic function ofZ.There existsρ∈(0,1)such thatϕ(t )=O(ρ^|t|1/H) (|t| →

∞). Consequently, the probability distribution of Z possesses an infinitely differentiable bounded density, and E(|Z|^−γ) <∞for allγ∈(0,1).

Proof. The caseH=1 is obvious. Suppose thatH∈(1/2,1). The probability distribution ofZ cannot be a Dirac mass, becauseE(Z)=1 and

Z=2^−H(0)Z(0)+2^−H(1)Z(1), (2.2)

with the same independence and equidistribution properties as in (1.7). So there existsα >0 andγ <1 such that sup_{t,|t|∈[α,2}Hα]|ϕ(t )| ≤γ. Now, using the fact that

ϕ(t )= p⁺_Hϕ

2^−Ht +p⁻_Hϕ

−2^−Ht ²,

we obtain by induction that sup_t,|t|∈[2kHα,2^(k+1)H]α]|ϕ(t )| ≤γ^2k (∀k≥0). Since|t|^1/H ≤2α^1/H2^k for |t| ∈ [2^kHα, 2^(k+1)Hα], the conclusion follows withρ=γ^1/2α1/H.

The rate of decay ofϕ at∞yields the conclusion regarding the probability distribution ofZ and the moments

of|Z|⁻¹.

Proof of Theorem1.1: the convergence properties of(B_n)_n≥1and the global Hölder continuity of the limit process Letα∈(0, H ). It follows from Lemma 2.2 that with probability 1, there exists δ >0 and C >0 such that for all (t, s)∈ [0,1]² such that|t−s| ≤δ we have sup_n≥1|B_n(t )−B_n(s)| ≤C|t−s|^α (see for instance the proof of the Kolmogorov–Centsov theorem in [20]). Since the sequence (B_n)_n≥1 converges almost surely on the set of dyadic numbers of[0,1]which is dense in[0,1], this implies that, with probability 1,(B_n)_n≥1converges uniformly to a limit Bwhich isα-Hölder continuous. To see that the convergence holds inL^qnorm for allq≥1, it is enough to show that the sequence(E(sup_1≤p≤nB_p^q_∞))_n≥1is bounded for all integerq≥2. We show that it is true forq=2 and leave the reader verify by induction that it is true forq≥2. Forn≥1, define

Zn= sup

1≤p≤nBp_∞ and Zn(k)= sup

1≤p≤n

Bp(k)

∞, k∈ {0,1}.

Due to (1.3) we have forn≥2 Z_n≤max

2^−HZ_n−1(0),2^−HZ_n−1(1)+ sup

1≤p≤n

B_p(1/2).

Thus, if we denote sup_1≤p≤n|Bp(1/2)|byMnwe have EZ²_n ≤E

2^−2HZ_n−1(0)²+2^−2HZ_n−1(1)²+2Z_n−1(1)Mn+M_n²

≤2^1−2HEZ_n−² ₁ +2EZ²_n−1 ^1/2Mn2+ Mn²2.

Lemma 2.1 shows that(B_p(1/2))p≥1is a martingale bounded inL²norm, so(M_n2)_n≥1is bounded. Consequently, there existsC >0 such that

∀n≥1, EZ_n² ≤f

EZ²_n−1

withf (x)=2^1−2Hx+C√

x+C. (2.3)

Since 2^1−2H<1, there existsx₀>0 such thatf (x) < xfor allx > x₀. This fact together with (2.3) yieldsE(Z_n²)≤ max(x0, f (E(Z₁²)))for alln≥2.

Proof of Theorem1.1: the properties1–3

1. This is an immediate consequence of (1.3).

2. The global Hölder regularity property has already been established. To obtain the pointwise Hölder exponent we use an approach similar to that used for the Brownian motion in [10] (see also [20]).

Fixε >0 and letO be the set of points ω∈Ω such that B_n converges uniformly asn→ ∞and the limitB possesses points at which the pointwise Hölder exponent is at leastH+ε. We show thatOis included in a set of null probability.

We fix an integerK >4/ε and denote by n_K the smallest integern such that K2⁻ⁿ≤1. For t ∈ [0,1] and n≥n_K, consider S_n^K(t )a subset of [0,1] consisting ofK+1 consecutive dyadic numbers of generationn such thatt ∈ [minS_n^K(t ),maxS_n^K(t )]. Also denote byS^K_n(t )the set ofK consecutive dyadic intervals delimited by the elements ofS_n^K(t ). If the pointwise Hölder exponent att is larger than or equal to H+εthen forn large enough we have necessarily sup_s∈SK

n(t )|B(s)−B(t )| ≤(K2⁻ⁿ)^H+ε/2, so that sup_I∈SK

n(t )|ΔB(I )| ≤2(K2⁻ⁿ)^H+ε/2, where ΔB(I )stands for the increment ofB overI.

Now letS^K_n be the set consisting of allK-uple of consecutive dyadic intervals of generationn, and ifS∈S^K_n, denote the event{sup_I∈S|ΔB(I )| ≤2(K2⁻ⁿ)^H+ε/2}byES. The previous lines show that

O⊂O=

n≥nK

p≥n

S∈S^K_p

E_S.

By construction, ifS∈S^K_p,(|ΔB(I )|)I∈Sis equal to(2^−pH|YI|)I∈S, where theKrandom variablesYI are mutually independent and identically distributed withB(1). Consequently,P(E_S)depends only onKandpand

P(E_S)≤

PB(1)≤2K^H

K2^{−p ε/2 K}

≤

2K^{H K/2}K^Kε/42^−pKε/4

EB(1)^{−1/2 K},

whereE(|B(1)|^−1/2) <∞due to Lemma 2.3. Since the cardinality ofS^K_p is less than 2^p, this yieldsP(

S∈S^K_pES)= O(2^p2^−pKε/4). Our choice forKimplies that the series

p≥nKP(

S∈S^K_pE_S)converges, henceP(O)=0.

3. Let us introduce additional notations. Ifw∈^∗andJ=Iwthen we define(J ):=(w)=_|w|

k=1(w1· · ·wk).

We denote byΓ the graph{(t, B(t )): t∈ [0,1]} ofB. We recall that the Hausdorff dimension of a subset ofR²is always smaller than of equal to its box dimension.

At first, sinceB isα-Hölder continuous for allα < H, 2−H is an upper bound for the box dimension ofΓ (see [13], Chapter 11).

To find the sharp lower bound 2−H for the Hausdorff dimension ofΓ we show that, with probability 1, the measure on this graph obtained as the image of the Lebesgue measure restricted to[0,1]by the mappingt→(t, B(t )) has a finite energy with respect to the Riesz Kernelu∈R²\ {0} → u^−γ for allγ <2−H (see [13], Chapter 4.3 and 11 for details about this kind of approach). This property holds if we show that for allγ <2−H we have

[0,1]²E

|t−s|²+B(t )−B(s)^{2 −γ /2} dtds <∞.

IfI is a closed subinterval of[0,1], we denote byG(I )the set of closed dyadic intervals of maximal length included inI, and thenmI=min

J∈G(I )JandMI=max

J∈G(I )J.

Let 0< s < t <1 be two non dyadic numbers. We define two sequences(s_p)_p≥0and(t_p)_p≥0as follows. Lets₀= m_[s,t]andt₀=M_[s,t]. Then let define inductively(s_p)_p≥1and(t_p)_p≥1as follows:s_p=m_[s,sp−1]andt_p=M_[tp−1,t]. Let us denote byCthe collection of intervals consisting of[s0, t0]and all the intervals[sp, sp−1]and[tp−1, tp],p≥1.

Every intervalI∈Cis the union of at most two intervals of the same generationnI, the elements ofG(I ), and ΔB(I )=

J∈G(I )

ΔB(J )=

J∈G(I )

(J )2^−nIHYJ,

whereΔB(J )andY (J )have been introduced in the discussion regarding the pointwise exponents. By construction, we have minI∈Cn_I=n_[s0,t0] and (t−s)/3≤2^−n[s0,t0] ≤(t−s). Also, all the random variablesY_J are mutually independent and independent ofT_C=σ ((J ): J∈G(I ), I∈C). Now, we write

B(t )−B(s)=2^−n[s0,t0]H

J∈G([s0,t0])

(J )Y_J+Z(s, s₀)+Z(t₀, t )

,

where

Z(s, s₀)=limp→∞

0≤k≤p2^{(n[s0,t0]−n[sk+1,sk])H}

J∈G([sk+1,sk])(J )Y_J, Z(t₀, t )=limp→∞

0≤k≤p2^{(n[s0,t0]−n[tk ,tk+1])H}

J∈G([t_k,t_k+1])(J )Y_J. LetZ(t, s)=

J∈G([s₀,t₀])(J )Y_J +Z(s, s0)+Z(t0, t )and fixJ0∈G([s0, t0]). Conditionally onT_C,Z(t, s)is the sum of±Y (J0)plus a random variableUindependent ofY (J0). Consequently, the probability distribution ofZ(t, s) conditionally onT_C possesses a densityf_t,s andf_t,sL¹≤ ϕL¹, whereϕ is the characteristic function ofY (J0) studied in Lemma 2.3.

Thus, forγ <2−Hwe have

E

|t−s|²+B(t )−B(s)^{2 −γ /2}|T_C =

R

f_t,s(u)

(|t−s|²+2^{−2n[s0,t0]H}u²)^{γ /2}du

≤

R

ft,s(u)

(|t−s|²+3^−2H(t−s)^2Hu²)^{γ /2}du

= |t−s|^1−H−γ

R

f_t,s(|t−s|^1−Hv) (1+3^−2Hv²)^{γ /2} dv.

The functionf_t,s is bounded independently oft, sandT_Csince it is bounded byf_t,sL¹ and we just saw that this number is bounded byϕ_L¹. Thus,

E

|t−s|²+B(t )−B(s)^{2 −γ /2} ≤ ϕL¹|t−s|^1−H−γ

R

dv (1+3^−2Hv²)^{γ /2}.

This yields the conclusion. Notice that the fact that the distribution of the increment ofBover[0,1], namelyZ, has a density plays a crucial role in this proof, as the same kind of property is a powerful tool in finding a lower bound for the Hausdorff dimension of the graphs of fractional Brownian motions, symmetric Lévy processes of indexα∈(1,2) and certain Weierstrass functions with random phases (see [13,16]).

3. Proof of Theorem 1.3

The caseH= −∞has been discussed in Remark 1.3. We fixH∈(−∞,1/2].

Lemma 3.1. The sequence(Xn(1))n≥1converges in law to the standard normal distribution asntends to∞.